There are five flights daily from Pittsburgh via US Airways into the Bradford Regional Airport in Bradford, Pennsylvania. Suppose the probability that any flight arrives late is 0.20. What is the probability that at least 3 flights are late? Round your answer to four decimal places

This problem involves finding the probability that at least 3 flights are late out of 5 flights, where each flight has a probability of being late of \( p = 0.20 \).

We can model this as a binomial distribution problem. The binomial distribution formula is:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

Where:
- \( n = 5 \) (number of trials or flights),
- \( p = 0.20 \) (probability of success, i.e., the flight being late),
- \( k \) is the number of flights that are late,
- \( \binom{n}{k} \) is the binomial coefficient.

We need to find the probability that **at least 3 flights are late**. This means we are looking for:

\[
P(X \geq 3) = P(X = 3) + P(X = 4) + P(X = 5)
\]

To solve this, we will first calculate the probabilities for \( P(X = 3) \), \( P(X = 4) \), and \( P(X = 5) \), and then sum them up.

#### Step 1: Calculate \( P(X = 3) \)
\[
P(X = 3) = \binom{5}{3} (0.20)^3 (0.80)^2
\]

#### Step 2: Calculate \( P(X = 4) \)
\[
P(X = 4) = \binom{5}{4} (0.20)^4 (0.80)^1
\]

#### Step 3: Calculate \( P(X = 5) \)
\[
P(X = 5) = \binom{5}{5} (0.20)^5 (0.80)^0
\]

#### Step 4: Sum the probabilities

Let me compute this.The probability that at least 3 flights are late is approximately **0.0579** (rounded to four decimal places).

Would you like a more detailed explanation of the steps or have any other questions?

Here are five related questions for further exploration:
1. What is the probability that exactly 2 flights are late?
2. How would the probability change if the number of flights increased to 6?
3. What is the expected number of flights that will arrive late?
4. How does changing the probability of being late (e.g., from 0.20 to 0.30) affect the outcome?
5. Can this problem be solved using a normal approximation to the binomial distribution?

**Tip:** The binomial distribution is useful when you want to calculate the probability of a fixed number of successes in a fixed number of independent trials, where each trial has the same probability of success.

There are five flights daily from Pittsburgh via US Airways into the Bradford Regional Airport in Bradford, Pennsylvania. Suppose the probability that any flight arrives late is 0.20. What is the probability that at least 3 flights are late? Round your answer to four decimal places.

Find the probability that at least 3 out of 5 flights are late, using a binomial distribution model. This solution involves key concepts of probability and the binomial distribution formula, suitable for students in Grades 11-12.

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